Geology Reference
In-Depth Information
Fig. 6.3
Tot al energy of a
prolate spheroidal SD
grain, magnetized in the
direction of the easy axis
(¥
0.4
[Nm x 10
-28
]
U
max
0.3
0,
dashed line
)asa
function of the rotation
angle ™. The curves have
been traced assuming
V
D
U
2
U
T
U
D
0.2
10
22
m
3
,
D
1.25
M
S
D
1A/m,
H
D
0.05 A/m,
U
0.1
1
and
X
/
Z
0.8).
In this example, the total
energy maximum is
attained for ™
D
0.60(
e
D
U
θ [°deg]
0.0
106
ı
D
θ
0
45
90
135
180
0
−
0.1
Similarly, to have a rotation from state 2 to
state 1, the thermal energy must be in excess of
U
21
D
U
max
-
U
2
. Insertion of (
6.14
)in(
6.11
)
gives:
where
K
12
and
K
21
are, respectively, the proba-
bility of a transition from state 1 (™
D
0
ı
) to state
2(™
D
180
ı
) and from state 2 to state 1. At any
time, the average magnetization is given by:
M.t/
D
ŒN
n.t/M
S
C
n.t/M
S
N
H
2
H
c
C
1
2
0
VM
S
U
max
D
0
VM
S
D
z
C
.D
x
D
z
/
1
2n.t/
N
N
H
2
H
c
D
M
S
(6.19)
(6.16)
In terms of
M
,Eq.(
6.18
) assumes the form:
Therefore, using (
6.15
) we see that the two
energy barriers for state transitions are given by:
M.t/
C
KM.t/
D
.K
21
K
12
/M
S
(6.20)
2
0
VM
S
H
c
1
C
2
1
H
H
c
where:
U
12
D
1
K
12
C
K
21
2
0
VM
S
H
c
1
2
K
D
(6.21)
1
H
H
c
U
21
D
(6.17)
The solution to Eq. (
6.20
) is immediate:
Clearly, for
H
> 0, the transition from state 1
to state 2 is more difficult than the reverse one.
Therefore, the number of grains,
n
, in state 1
increases progressively until the thermal equilib-
rium is attained. A kinetic equation that describes
this process can be obtained assuming that the
number of transitions per unit time from one state
to another is proportional to the corresponding
population of grains in the initial state. Therefore,
C
M
eq
1
e
t=
(6.22)
M.t/
D
M
0
e
t=
where
M
0
D
M
(0) and
M
eq
D
(
K
21
-
K
12
)
M
S
/
K
is
the limit equilibrium state for
t
!1
. A direct
formula for
M
eq
is expression (
6.1
). The quantity
£ is called
relaxation time
. It can be calculated
from the probabilities of state transitions
K
12
and
K
21
using (
6.21
). These quantities obey to an
Arrhenius equation for the temperature depen-
dence of transition rates, with thermal activation
P
n.t/
D
K
21
ŒN
n.t/
K
12
n.t/
(6.18)
